# Deductive system for natural language

The formal language of maths use the first order logic to deduce a theorem from a hypothesis. But, in the natural language, the subyacent logic can´t be first order logic. In natural languages all mathematicals are right with the existence of a standard model for arithmetic. The term "standard" can´t be translated to logic of first orden, and it implies, for example, we have infinite true phormulas for the natural numbers that can´t be deduced from Peano Axioms and any axioms in first order logic. What is the subyacent logic for natural languages?